5 COMPACT OPERATORS IN HUBERT SPACES 337

(b) Suppose E is a Hilbert space, and u is a compact self-adjoint operator. If u(x) =£ 0,
show that un(x) =£ 0 for any integer n > 0, and that the sequence of positive numbers
an = \\un+1(x) \\l\\if(x)\\ is increasing and tends to a limit, which is equal to the absolute
value of an eigenvalue of u. Characterize that eigenvalue in terms of the canonical
decomposition of x; when does the sequence of vectors u"(x)J \\un(x)\\ have a limit in E?
(Use (11.5.7)0

11. Let u be a compact self-adjoint operator in a complex Hilbert space E, and let/be a
complex valued function defined and continuous in the spectrum sp(w). Show that
there is a unique continuous operator v such that (with the notations of (11.5.7)), the
restriction of v to E(^k) (resp. E(vk), E(0)) is the homothetic mapping y ->/(/UK)J>
(resp. y-*-f(yk)y9y-*'Q)' This operator is written/(w); one has (/(«))* =/(«). If g is
a second function continuous in sp(#), and h—f+g (resp. h=fg), then h(u) =
f(u) 4- g(u) (resp. h(u) —f(u)g(u)). In order that/(z/) be self-adjoint (resp. positive and
self-adjoint), it is necessary and sufficient that /(£) be real in sp(u) (resp. /(£) ^ 0 in
sp(#)); in order that /(«) be compact, it is necessary and sufficient that/(0) = 0.

12. Let u be a compact positive hermitian operator in a complex Hilbert space E. Show
that there exists a unique compact positive hermitian operator u in E such that v2 = u\
v is called the square root of u.

13. Let E be a separable complex Hilbert space, (£„)«> i a Hilbert basis of E. Let
u be the compact operator in E defined by u(ei) = 0, u(en) — en~i/n for n > 1. Show
that there exists no continuous operator z; in E such that v2 ~ u. (Observe first that
H = «*(£) is a closed hyperplane orthogonal to ei9 and that it is contained in
H' = t?*(E); as H' is orthogonal to Xi = v(ei), conclude that necessarily x1==0;
next consider x2 = v(e2), and observe that u(v(e2)) = 0, hence necessarily x2 = Aei,
where A is a scalar; but this implies x2 = 0, hence u(e2) = 0, a contradiction.)

14. Let E be a separable complex Hilbert space, (en)n^0 a Hilbert basis, u the com-
pact positive hermitian operator in E defined by u(e0) = 0, u(en) = en/n for n 5? 1.

00

The point a = ]T (<?n//z) does not belong to w(E). Let E0 be the dense subspace of E

n=l

which is the direct sum of u(E) and of the one-dimensional subspace C(e0 -f a).
Show that the restriction v of u to EO is a compact positive hermitian operator which is
nondegenerate, although its continuous extension v = u to E is degenerate; further-
more, in the canonical decomposition (11.5.8) of the vector <?0 + # e EO , the summands
do not all belong to E0.

15. (a) Let U be a compact operator in a complex Hilbert space E, and denote by R and
L the respective square roots (Problem 12) of the compact positive herrnitian operators
U*U and UU*9 respectively. Show that there exists a unique continuous operator V
in E, whose restriction to F = R(E) is an isometry onto U(E), whose restriction to the
orthogonal supplement F' to F is 0, and which is such that £/= VR (observe that
\\Ux\\ = \\Rx\\ for each x e E). Prove that R = V*U = RV*V, and L = VRV*.

(b) Let (aM) the full sequence of strictly positive eigenvalues of R, and (an) a corre-
sponding orthonormal system (Problem 8). If bn — Vant show that (bn) is an ortho-
normal system, and that, for any x e E, Ux =~^<Xr,(x\an)bHy where the series on the

n «

right-hand side is convergent (if Rn x = V ak(x \ ak)ak, show, using the proof of

*=i

(11.5.7), that lim ||/?-7?J| = 0, and apply (a)). Deduce from that result that (<*„)

n-»-oo

is also the full sequence of strictly positive eigenvalues of L, and that (bn) is a cor-
responding orthonormal system. The sequence (an) is also called the full sequence of
singular values of U.\\un(x)\\2 < \\if~l(x)\\ - ||MW+I(JC)|| (use Cauchy-